## Algebra and Tiling: Homomorphisms in the Service of GeometryOften questions about tiling space or a polygon lead to questions concerning algebra. For instance, tiling by cubes raises questions about finite abelian groups. Tiling by triangles of equal areas soon involves Sperner's lemma from topology and valuations from algebra. The first six chapters of Algebra and Tiling form a self-contained treatment of these topics, beginning with Minkowski's conjecture about lattice tiling of Euclidean space by unit cubes, and concluding with Laczkowicz's recent work on tiling by similar triangles. The concluding chapter presents a simplified version of Rédei's theorem on finite abelian groups. Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper level algebra courses, but teachers, researchers and professional mathematicians will find the book equally appealing. |

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### Contents

Chapter 2 | 35 |

Chapter 3 | 54 |

Chapter 4 | 85 |

Chapter 5 | 105 |

Chapter 6 | 126 |

Appendix C | 143 |

Chapter 7 | 155 |

Epilog | 187 |

Formal Sums | 197 |

### Common terms and phrases

algebraic angles assume basis brick chapter character of G cluster coefficients complete complex numbers Consider convex convex polygon coordinates coset cotangents counterclockwise cyclic group cyclic subsets cyclotomic cyclotomic polynomial defined denote density dissection divides edge equal areas equidissection equivalent exact sequence Exercise 12 factorization G factorization of G Figure finite abelian group finite number geometry group G group of order group ring Hajos Hajos's theorem Hence homomorphism identity element least Let G Math Minkowski's conjecture modulo multiple n-dimensional n-space n)-cross n)-semicross nonzero normed factorization nth root odd number p-group polygon positive integer prime number prime orders primitive pth root problem proof of Redei's Prove quadratic form quotient group real numbers rectangle Redei's theorem relatively prime replacement principle root of unity semicross Show Sperner's Lemma splits G splitting set subgroup of G Suppose translates triangles of equal unit cubes valuation vectors vertices Z-lattice packing zero polynomial